Multiply the following complex numbers, marked as blue dots on the graph: $[\cos(\frac{1}{2}\pi) + i \sin(\frac{1}{2}\pi)] \cdot [2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))]$ (Your current answer will be plotted in orange.)
Solution: Multiplying complex numbers in polar forms can be done by multiplying the lengths and adding the angles. The first number ( $\cos(\frac{1}{2}\pi) + i \sin(\frac{1}{2}\pi)$ ) has angle $\frac{1}{2}\pi$ and radius $1$ The second number ( $2(\cos(\frac{1}{4}\pi) + i \sin(\frac{1}{4}\pi))$ ) has angle $\frac{1}{4}\pi$ and radius $2$ The radius of the result will be $1 \cdot 2$ , which is $2$ The angle of the result is $\frac{1}{2}\pi + \frac{1}{4}\pi = \frac{3}{4}\pi$ The radius of the result is $2$ and the angle of the result is $\frac{3}{4}\pi$.